08/11/22 18:43:45
>>40
a, b ∈ K, f, g ∈ F(N) と n ∈ N に対して
D(a f + b g)(n) = (a f + b g)(n+1) - (a f + b g)(n) = 中略 = a (D f)(n) + b (D g)(n)
なので D は線型。同様に
D'(a f + b g)(n) = (a f + b g)(1) + ... + (a f + b g)(n) = 中略 = a (D' f)(n) + b (D' g)(n)
なので D' も線型
さらに任意の n ∈ N に対して
(D D' f)(n) = (D (D' f))(n) = (D' f)(n+1) - D' f)(n) = 中略 = f(n+1)
同様に
(D' D f)(n) = (D' (D f))(n) = (D f)(1) + ... + (D f)(n) = 中略 = f(n+1) - f(n)
中略の部分は本当にただの計算